Difference between revisions of "Thermodynamics of Solid and Fluid Surfaces"
(→Plane Fluid Interfaces  Area Work Term) 

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Entry by [[Emily Redston]], AP 226, Spring 2012  Entry by [[Emily Redston]], AP 226, Spring 2012  
−  
−  
==Reference==  ==Reference==  
Line 7:  Line 5:  
==Keywords==  ==Keywords==  
−  +  [[thermodynamics]], [[interfaces]], [[excess free energy]], [[Phase Rule]], [[GibbsDuhem]]  
==Introduction==  ==Introduction==  
−  In this paper, Cahn provides a slightly different take on Gibbs' original development of the thermodynamics of fluid and solid surfaces. His goal was to develop a mathematical system that was physically equivalent to Gibbs' formulation while being more easily applicable. The key change he made is that he concentrated on the system rather than defining excess quantities by comparing the actual system to a hypothetical system not containing the surface. This avoids problems with the identification of the location of the dividing surface.  +  In this paper, Cahn provides a slightly different take on Gibbs' original development of the thermodynamics of fluid and solid surfaces. His goal was to develop a mathematical system that was physically equivalent to Gibbs' formulation while being more easily applicable. The key change he made is that he concentrated on the system itself rather than defining excess quantities by comparing the actual system to a hypothetical system not containing the surface. This avoids problems with the identification of the location of the dividing surface. 
+  
+  In his paper, Cahn uses a shorthand notation for the matrices, but I chose to write it out fully here since I find it a little clearer.  
==Plane Fluid Interfaces  Area Work Term==  ==Plane Fluid Interfaces  Area Work Term==  
−  For surfaces between fluids, Gibbs defined surface tension  +  For surfaces between fluids, Gibbs defined surface tension <math>\sigma</math> in a work term. The quantity <math>\sigma</math> is taken as a force per unit length of surface perimeter. When a portion of the perimeter moves an infinitesimal distance in the plane of the surface, the area change <math>dA</math> is the product of perimeter length and distance moved. Thus <math>\sigma dA</math> is a forcetimesdistance work term, and it appears in the combined first and second laws of thermodynamics as follows: 
+  
+  <center><math>dU = TdS  PdV + \displaystyle \sum_{i} \mu_i dN_i + \sigma dA\ [1]</math> </center>  
+  
+  Strictly speaking, <math>\sigma</math> is defined as the change in internal energy when the area is reversibly increased at constant entropy and volume in a closed system.  
+  
+  Gibbs went on to show that, in equilibrium, temperature and chemical potentials are constant everywhere within the system. For the special case of plane boundaries in equilibrium, pressure is also constant. For a system containing a plane surface, Equation 1 can be integrated to give:  
+  
+  <center><math>U = TS  PV + \displaystyle \sum_{i} \mu_i N_i + \sigma A\ [2] </math></center>  
+  
+  This can be rearranged to give <math>\sigma</math>, the excess free energy due to the presence of the surface:  
+  
+  <center><math> \sigma = (1/A) [U  TS + PV  \displaystyle \sum_{i} \mu_i N_i]\ [3]</math></center>  
+  
+  Here <math>U  TS + PV</math> is the Gibbs free energy of the system with the interface, and <math>\displaystyle \sum_{i} \mu_i N_i</math> is the Gibbs free energy without the interface.  
+  
+  Equations [2] and [3] pertain to the entire system. Because most of this system is occupied by homogeneous phases (which we are not so interested in), it is convenient to subtract out their contributions. For homogeneous phases <math>\alpha</math> and <math>\beta</math>, we have:  
+  
+  <center><math> 0 = [U^{\alpha}  TS^{\alpha} + PV^{\alpha}  \displaystyle \sum_{i} \mu_i N_i^{\alpha}]\ [4]</math></center>  
+  
+  <center><math> 0 = [U^{\beta}  TS^{\beta} + PV^{\beta}  \displaystyle \sum_{i} \mu_i N_i^{\beta}]\ [5]</math></center>  
+  
+  It is apparent that the homogeneous phases contribute nothing to the RHS of Equation 3, and Equation 3 continues to hold if we remove from our consideration as much of the homogeneous phases as we wish. What is left then is a layer which is thick enough to extend into both homogeneous phases. The location of the bounds of this layer parallel to the surface do not matter, except that they must reach far enough into each phase such that there is no longer any influence of the surface. Equations 4 and 5 hold for any incremental addition to the thickness of this layer. If we define the layer content per unit surface area of extensive quantities and denote them by the symbols [U] [S] [V] and [<math>N_i</math>], then <math>\sigma</math>, which was defined as a force per unit length in a work term, is identically equal to the Gibbs free energy of forming this layer containing the surface:  
+  
+  <center><math> \sigma = [U]  T[S] + P[V]  \displaystyle \sum_{i} \mu_i [N_i]\ [6]</math></center>  
+  
+  It is important to note that the layer quantities are not excess quantities; <i>they are the actual amounts present in such a layer</i>. As with Gibbs' excess quantities, the layer quantities are dependent on the placement of the layer bounds. In other words, [U], [S], and [<math>N_i</math>] can take on a wide range of values depending on the arbitrary decision of where in the homogeneous phases one decides to place the layer bounds. Thus these quantities by themselves can never be important. However certain combinations, as in Equation 6, are independent of such decisions, and this invariance is why we are interested in the surface free energy <math>\sigma</math>.  
+  
+  By combining Equations 1 and 2 and making use of the definition of layer quantities, Cahn obtains a version of the Gibbs absorption equation:  
+  
+  <center><math>d\sigma = [S]dT + [V]dP  \displaystyle \sum_{i} [N_i]d\mu_i\ [7]</math></center>  
+  
+  From this equation, you can derive:  
+  
+  <center><math>({\partial \sigma\over\partial T})_{P,\mu_i} = [S]\ [8]</math></center>  
+  
+  However, Equation 8 is not particularly useful because it is arbitrary. The RHS is one of the meaningless layer quantities, while the LHS is an impossible variation. Because the surface is in contact with the phases, it is impossible to vary temperature while holding pressure and chemical potential constant; it would violate the phase rule.  
+  
+  In the derivation of Gibbs' phase rule, the GibbsDuhem equations play an important role in restricting the allowed variation. For each phase there is one GibbsDuhem equation and one lost degree of freedom:  
+  
+  <center><math> 0 = S^{\alpha}dT + V^{\alpha}dP  \displaystyle \sum_{i} N_i^{\alpha}d\mu_i\ [9]</math></center>  
+  
+  <center><math> 0 = S^{\beta}dT + V^{\beta}dP  \displaystyle \sum_{i} N_i^{\beta}d\mu_i\ [10]</math></center>  
+  
+  Cahn uses these two equations with Equation 7 to solve for how three members of the set {d<math>\sigma</math>, dT, dP, d<math>\mu_i</math>} behave when we control the remainder. Applying [http://en.wikipedia.org/wiki/Cramer%27s_rule Cramer's rule] to these equations, Cahn obtains:  
+  <center><math>  
+  \begin{bmatrix}  
+  d\sigma & [X] & [Y] \\  
+  0 & X^{\alpha} & Y^{\alpha}\\  
+  0 & X^{\beta} & Y^{\beta}  
+  \end{bmatrix}  
+  =  
+    
+  \begin{bmatrix}  
+  [S] & [X] & [Y] \\  
+  S^{\alpha} & X^{\alpha} & Y^{\alpha}\\  
+  S^{\beta} & X^{\beta} & Y^{\beta}  
+  \end{bmatrix}  
+  dT  
+  +  
+  \begin{bmatrix}  
+  [V] & [X] & [Y] \\  
+  V^{\alpha} & X^{\alpha} & Y^{\alpha}\\  
+  V^{\beta} & X^{\beta} & Y^{\beta}  
+  \end{bmatrix}  
+  dP  
+    
+  \displaystyle \sum_{i}  
+  \begin{bmatrix}  
+  [N_i] & [X] & [Y] \\  
+  N_i^{\alpha} & X^{\alpha} & Y^{\alpha}\\  
+  N_i^{\beta} & X^{\beta} & Y^{\beta}  
+  \end{bmatrix}  
+  d\mu_i\ [11]  
+  </math></center>  
+  
+  where X and Y are any two distinct member of the set {S, V, <math>N_i</math>}.  
+  
+  <div style="border:3px solid #BCD4E6; {{Round corners}}; margin: 5px;">  
+  In general, we can define a quantity [Z/XY] that is the excess of Z in the layer over what would be in a comparison system of two phases containing the same amount of X and Y. In other words, it is the difference in the amount of Z between unit area of the layer and two portions of the homogeneous phase having the same total amount of X and Y as the layer. Here Z can be S,V, etc. <b>Proof:</b> Let the layer content be [Z], [X] and [Y] and let the content of a given amount of <math>\alpha</math> and <math>\beta</math> phase be <math>Z^{\alpha}, X^{\alpha},...., Y^{\beta}</math>. Take <math>k^{\alpha}</math> and <math>k^{\beta}</math> units of <math>\alpha</math> and <math>\beta</math> phase such that:  
+  
+  <center><math>[X] = k^{\alpha}X^{\alpha} + k^{\beta}X^{\beta}</math></center>  
+  
+  <center><math>[Y] = k^{\alpha}Y^{\alpha} + k^{\beta}Y^{\beta}</math></center>  
+  
+  Then the excess of Z in the layer, <math>\Delta Z</math> is given by:  
+  
+  <center><math>[Z] = k^{\alpha}Z^{\alpha} + k^{\beta}Z^{\beta} + \Delta Z</math></center>  
+  
+  Solving for <math>\Delta Z</math> by eliminating <math>k^{\alpha}</math> and <math>k^{\beta}</math>, once again using Cramer's rule, gives:  
+  
+  <center><math> \Delta Z = [Z/XY] =  
+  
+  {\begin{bmatrix}  
+  [Z] & [X] & [Y] \\  
+  Z^{\alpha} & X^{\alpha} & Y^{\alpha}\\  
+  Z^{\beta} & X^{\beta} & Y^{\beta}  
+  \end{bmatrix}  
+  
+  \over  
+  
+  \begin{bmatrix}  
+  X^{\alpha} & Y^{\alpha}\\  
+  X^{\beta} & Y^{\beta}  
+  \end{bmatrix}}  
+  </math></center>  
+  
+  This definition of excess quantities is equivalent to those Gibbs defined with one important exception. In Gibbs' development, either X or Y always had to be the volume. Thus Cahn's method offers a little more flexibility  
+  </div>  
+  
+  ==Examples==  
+  
+  <b>1. Single component, Two Phases</b>  
+  
+  Let's consider a single component solution in contact with its vapor. According to Gibbs phase rule, we only have one degree of freedom. So if we take our intensive variable controlled by experiment to be the temperature, then our pressure is fixed. This means that we choose V and N (where <math>N=N_1</math>) for X and Y in Equation 11. Thus we are looking at the excess entropy over a layer of the same V and N:  
+  
+  <center><math>d\sigma =  
+  {\begin{bmatrix}  
+  [S] & [V] & [N] \\  
+  S^{\alpha} & V^{\alpha} & N^{\alpha}\\  
+  S^{\beta} & V^{\beta} & N^{\beta}  
+  \end{bmatrix}  
+  
+  \over  
+  
+  \begin{bmatrix}  
+  V^{\alpha} & N^{\alpha}\\  
+  V^{\beta} & N^{\beta}  
+  \end{bmatrix}}dT\ [12]  
+  </math></center>  
+  
+  <b>2. Two components, Two Phases</b>  
+  
+  Let's consider a binary solution in contact with its vapor. According to Gibbs phase rule, we now have two degrees of freedom. Imagine we take the intensive variables controlled by experiment to be the temperature and the chemical potential of component 2. The pressure and the chemical potential of component 1 are fixed, or dP and d<math>\mu_1</math> are zero. This means that we choose V and <math>N_1</math> for X and Y in Equation 11. Thus we are looking at the excess entropy and interface excess of <math>N_2</math> at zero excess of <math>N_1</math> and V:  
+  
+  <center><math>d\sigma =  
+  {\begin{bmatrix}  
+  [S] & [V] & [N] \\  
+  S^{\alpha} & V^{\alpha} & {N_1}^{\alpha}\\  
+  S^{\beta} & V^{\beta} & {N_1}^{\beta}  
+  \end{bmatrix}  
+  
+  \over  
+  
+  \begin{bmatrix}  
+  V^{\alpha} & {N_1}^{\alpha}\\  
+  V^{\beta} & {N_1}^{\beta}  
+  \end{bmatrix}}dT  
+  
+    
+  
+  {\begin{bmatrix}  
+  [{N_2}] & [V] & [N] \\  
+  {N_2}^{\alpha} & V^{\alpha} & {N_1}^{\alpha}\\  
+  {N_2}^{\beta} & V^{\beta} & {N_1}^{\beta}  
+  \end{bmatrix}  
+  
+  \over  
+  
+  \begin{bmatrix}  
+  V^{\alpha} & {N_1}^{\alpha}\\  
+  V^{\beta} & {N_1}^{\beta}  
+  \end{bmatrix}}dT\ [13]  
+  </math></center>  
+  
+  If we consider an isotherm (dT=0):  
+  
+  <center><math> d\sigma= \Gamma_2 d\mu_2</math></center>  
+  
+  By further looking at a dilute solution (<math>\mu_2={\mu_2}^{\circ} + RTln{c_2} \longrightarrow d\mu_2 = RT {d{c_2} \over {c_2}}</math>), we get <i>Gibbs absorption isotherm</i>:  
+  
+  
+  <center><math>{d\sigma \over d{c_2}} = RT{\Gamma_2 \over c_2}</math></center>  
+  
+  <b>2. Single Phase</b>  
+  
+  A common type of surface within solids is what may be called a singlephase interface. These are interfaces between regions of the same phase. Examples are grain boundaries, twin boundaries, stacking faults, and antiphase boundaries. For these systems, only one GibbsDuhem relation is applicable and, as a results, the Gibbs absorption equation may involve only one eliminated variable:  
+  
+  <center><math>  
+  \begin{bmatrix}  
+  d\sigma & [X]\\  
+  0 & {X}^{\alpha}  
+  \end{bmatrix}  
+  
+  =  
+    
+  \begin{bmatrix}  
+  [S] & [X]\\  
+  {S}^{\alpha}& {X}^{\alpha}  
+  \end{bmatrix}dT  
+  +  
+  \begin{bmatrix}  
+  [V] & [X]\\  
+  {V}^{\alpha}& {X}^{\alpha}  
+  \end{bmatrix}dP  
+    
+  \displaystyle \sum_{i}  
+  \begin{bmatrix}  
+  [N_i] & [X]\\  
+  {N_i}^{\alpha}& {X}^{\alpha}  
+  \end{bmatrix}d\mu_i  
+  
+  </math></center>  
−  +  If we pick <math>N_1</math> for X, the equation becomes:  
+  <center><math>  
+  d\sigma =  
+    
+  \begin{bmatrix}  
+  [S]  {{S}^{\alpha} \over {N_1}^{\alpha}}[{N_1}]\\  
+  \end{bmatrix}dT  
+  +  
+  \begin{bmatrix}  
+  [V]  {{V}^{\alpha} \over {N_1}^{\alpha}}[{N_1}]\\  
+  \end{bmatrix}dP  
+    
+  \displaystyle \sum_{i=2}  
+  \begin{bmatrix}  
+  [N_i]  {{N_i}^{\alpha} \over {N_1}^{\alpha}}[{N_1}]\\  
+  \end{bmatrix}d\mu_i  
−  <math></  +  </math></center> 
Latest revision as of 16:16, 19 April 2012
Entry by Emily Redston, AP 226, Spring 2012
Reference
Thermodynamics of Solid and Fluid Surfaces by J. W. Cahn. Segregation to Interfaces, ASM Seminar Series (1978) pp. 323.
Keywords
thermodynamics, interfaces, excess free energy, Phase Rule, GibbsDuhem
Introduction
In this paper, Cahn provides a slightly different take on Gibbs' original development of the thermodynamics of fluid and solid surfaces. His goal was to develop a mathematical system that was physically equivalent to Gibbs' formulation while being more easily applicable. The key change he made is that he concentrated on the system itself rather than defining excess quantities by comparing the actual system to a hypothetical system not containing the surface. This avoids problems with the identification of the location of the dividing surface.
In his paper, Cahn uses a shorthand notation for the matrices, but I chose to write it out fully here since I find it a little clearer.
Plane Fluid Interfaces  Area Work Term
For surfaces between fluids, Gibbs defined surface tension <math>\sigma</math> in a work term. The quantity <math>\sigma</math> is taken as a force per unit length of surface perimeter. When a portion of the perimeter moves an infinitesimal distance in the plane of the surface, the area change <math>dA</math> is the product of perimeter length and distance moved. Thus <math>\sigma dA</math> is a forcetimesdistance work term, and it appears in the combined first and second laws of thermodynamics as follows:
Strictly speaking, <math>\sigma</math> is defined as the change in internal energy when the area is reversibly increased at constant entropy and volume in a closed system.
Gibbs went on to show that, in equilibrium, temperature and chemical potentials are constant everywhere within the system. For the special case of plane boundaries in equilibrium, pressure is also constant. For a system containing a plane surface, Equation 1 can be integrated to give:
This can be rearranged to give <math>\sigma</math>, the excess free energy due to the presence of the surface:
Here <math>U  TS + PV</math> is the Gibbs free energy of the system with the interface, and <math>\displaystyle \sum_{i} \mu_i N_i</math> is the Gibbs free energy without the interface.
Equations [2] and [3] pertain to the entire system. Because most of this system is occupied by homogeneous phases (which we are not so interested in), it is convenient to subtract out their contributions. For homogeneous phases <math>\alpha</math> and <math>\beta</math>, we have:
It is apparent that the homogeneous phases contribute nothing to the RHS of Equation 3, and Equation 3 continues to hold if we remove from our consideration as much of the homogeneous phases as we wish. What is left then is a layer which is thick enough to extend into both homogeneous phases. The location of the bounds of this layer parallel to the surface do not matter, except that they must reach far enough into each phase such that there is no longer any influence of the surface. Equations 4 and 5 hold for any incremental addition to the thickness of this layer. If we define the layer content per unit surface area of extensive quantities and denote them by the symbols [U] [S] [V] and [<math>N_i</math>], then <math>\sigma</math>, which was defined as a force per unit length in a work term, is identically equal to the Gibbs free energy of forming this layer containing the surface:
It is important to note that the layer quantities are not excess quantities; they are the actual amounts present in such a layer. As with Gibbs' excess quantities, the layer quantities are dependent on the placement of the layer bounds. In other words, [U], [S], and [<math>N_i</math>] can take on a wide range of values depending on the arbitrary decision of where in the homogeneous phases one decides to place the layer bounds. Thus these quantities by themselves can never be important. However certain combinations, as in Equation 6, are independent of such decisions, and this invariance is why we are interested in the surface free energy <math>\sigma</math>.
By combining Equations 1 and 2 and making use of the definition of layer quantities, Cahn obtains a version of the Gibbs absorption equation:
From this equation, you can derive:
However, Equation 8 is not particularly useful because it is arbitrary. The RHS is one of the meaningless layer quantities, while the LHS is an impossible variation. Because the surface is in contact with the phases, it is impossible to vary temperature while holding pressure and chemical potential constant; it would violate the phase rule.
In the derivation of Gibbs' phase rule, the GibbsDuhem equations play an important role in restricting the allowed variation. For each phase there is one GibbsDuhem equation and one lost degree of freedom:
Cahn uses these two equations with Equation 7 to solve for how three members of the set {d<math>\sigma</math>, dT, dP, d<math>\mu_i</math>} behave when we control the remainder. Applying Cramer's rule to these equations, Cahn obtains:
\begin{bmatrix}
d\sigma & [X] & [Y] \\ 0 & X^{\alpha} & Y^{\alpha}\\ 0 & X^{\beta} & Y^{\beta}
\end{bmatrix} =  \begin{bmatrix}
[S] & [X] & [Y] \\ S^{\alpha} & X^{\alpha} & Y^{\alpha}\\ S^{\beta} & X^{\beta} & Y^{\beta}
\end{bmatrix} dT + \begin{bmatrix}
[V] & [X] & [Y] \\ V^{\alpha} & X^{\alpha} & Y^{\alpha}\\ V^{\beta} & X^{\beta} & Y^{\beta}
\end{bmatrix} dP  \displaystyle \sum_{i} \begin{bmatrix}
[N_i] & [X] & [Y] \\ N_i^{\alpha} & X^{\alpha} & Y^{\alpha}\\ N_i^{\beta} & X^{\beta} & Y^{\beta}
\end{bmatrix} d\mu_i\ [11]
</math>where X and Y are any two distinct member of the set {S, V, <math>N_i</math>}.
In general, we can define a quantity [Z/XY] that is the excess of Z in the layer over what would be in a comparison system of two phases containing the same amount of X and Y. In other words, it is the difference in the amount of Z between unit area of the layer and two portions of the homogeneous phase having the same total amount of X and Y as the layer. Here Z can be S,V, etc. Proof: Let the layer content be [Z], [X] and [Y] and let the content of a given amount of <math>\alpha</math> and <math>\beta</math> phase be <math>Z^{\alpha}, X^{\alpha},...., Y^{\beta}</math>. Take <math>k^{\alpha}</math> and <math>k^{\beta}</math> units of <math>\alpha</math> and <math>\beta</math> phase such that:
Then the excess of Z in the layer, <math>\Delta Z</math> is given by:
Solving for <math>\Delta Z</math> by eliminating <math>k^{\alpha}</math> and <math>k^{\beta}</math>, once again using Cramer's rule, gives:
{\begin{bmatrix}
[Z] & [X] & [Y] \\ Z^{\alpha} & X^{\alpha} & Y^{\alpha}\\ Z^{\beta} & X^{\beta} & Y^{\beta}
\end{bmatrix}
\over
\begin{bmatrix}
X^{\alpha} & Y^{\alpha}\\ X^{\beta} & Y^{\beta}
\end{bmatrix}}
</math>This definition of excess quantities is equivalent to those Gibbs defined with one important exception. In Gibbs' development, either X or Y always had to be the volume. Thus Cahn's method offers a little more flexibility
Examples
1. Single component, Two Phases
Let's consider a single component solution in contact with its vapor. According to Gibbs phase rule, we only have one degree of freedom. So if we take our intensive variable controlled by experiment to be the temperature, then our pressure is fixed. This means that we choose V and N (where <math>N=N_1</math>) for X and Y in Equation 11. Thus we are looking at the excess entropy over a layer of the same V and N:
{\begin{bmatrix}
[S] & [V] & [N] \\ S^{\alpha} & V^{\alpha} & N^{\alpha}\\ S^{\beta} & V^{\beta} & N^{\beta}
\end{bmatrix}
\over
\begin{bmatrix}
V^{\alpha} & N^{\alpha}\\ V^{\beta} & N^{\beta}
\end{bmatrix}}dT\ [12]
</math>2. Two components, Two Phases
Let's consider a binary solution in contact with its vapor. According to Gibbs phase rule, we now have two degrees of freedom. Imagine we take the intensive variables controlled by experiment to be the temperature and the chemical potential of component 2. The pressure and the chemical potential of component 1 are fixed, or dP and d<math>\mu_1</math> are zero. This means that we choose V and <math>N_1</math> for X and Y in Equation 11. Thus we are looking at the excess entropy and interface excess of <math>N_2</math> at zero excess of <math>N_1</math> and V:
{\begin{bmatrix}
[S] & [V] & [N] \\ S^{\alpha} & V^{\alpha} & {N_1}^{\alpha}\\ S^{\beta} & V^{\beta} & {N_1}^{\beta}
\end{bmatrix}
\over
\begin{bmatrix}
V^{\alpha} & {N_1}^{\alpha}\\ V^{\beta} & {N_1}^{\beta}
\end{bmatrix}}dT

{\begin{bmatrix}
[{N_2}] & [V] & [N] \\ {N_2}^{\alpha} & V^{\alpha} & {N_1}^{\alpha}\\ {N_2}^{\beta} & V^{\beta} & {N_1}^{\beta}
\end{bmatrix}
\over
\begin{bmatrix}
V^{\alpha} & {N_1}^{\alpha}\\ V^{\beta} & {N_1}^{\beta}
\end{bmatrix}}dT\ [13]
</math>If we consider an isotherm (dT=0):
By further looking at a dilute solution (<math>\mu_2={\mu_2}^{\circ} + RTln{c_2} \longrightarrow d\mu_2 = RT {d{c_2} \over {c_2}}</math>), we get Gibbs absorption isotherm:
2. Single Phase
A common type of surface within solids is what may be called a singlephase interface. These are interfaces between regions of the same phase. Examples are grain boundaries, twin boundaries, stacking faults, and antiphase boundaries. For these systems, only one GibbsDuhem relation is applicable and, as a results, the Gibbs absorption equation may involve only one eliminated variable:
\begin{bmatrix}
d\sigma & [X]\\ 0 & {X}^{\alpha}
\end{bmatrix}
=  \begin{bmatrix}
[S] & [X]\\ {S}^{\alpha}& {X}^{\alpha}
\end{bmatrix}dT + \begin{bmatrix}
[V] & [X]\\ {V}^{\alpha}& {X}^{\alpha}
\end{bmatrix}dP  \displaystyle \sum_{i} \begin{bmatrix}
[N_i] & [X]\\ {N_i}^{\alpha}& {X}^{\alpha}
\end{bmatrix}d\mu_i
</math>If we pick <math>N_1</math> for X, the equation becomes:
d\sigma =  \begin{bmatrix}
[S]  {{S}^{\alpha} \over {N_1}^{\alpha}}[{N_1}]\\
\end{bmatrix}dT + \begin{bmatrix}
[V]  {{V}^{\alpha} \over {N_1}^{\alpha}}[{N_1}]\\
\end{bmatrix}dP  \displaystyle \sum_{i=2} \begin{bmatrix}
[N_i]  {{N_i}^{\alpha} \over {N_1}^{\alpha}}[{N_1}]\\
\end{bmatrix}d\mu_i
</math>